An air-track glider is attached to a spring. The glider is...

An air-track glider is attached to a spring. The glider is pulled to the right and released from rest at $t=0$ s. It then oscillates with a period of $2.0 \mathrm{s}$ and a maximum speed of $40 \mathrm{cm} / \mathrm{s}$ a. What is the amplitude of the oscillation? b. What is the glider's position at $t=0.25 \mathrm{s} ?$

An air-track glider is attached to a spring. The glider is pulled to the right and released from rest at $t=0$ s. It then oscillates with a period of $2.0 \mathrm{s}$ and a maximum speed of $40 \mathrm{cm} / \mathrm{s}$
a. What is the amplitude of the oscillation?
b. What is the glider's position at $t=0.25 \mathrm{s} ?$

Key Concepts

Phase and Position Calculation

The position of an object in simple harmonic motion can be determined through the equation x(t) = A cos(?t + ?), where A is the amplitude, ? is the angular frequency, t is time, and ? is the phase constant. Choosing appropriate phase constants and plugging in the value of time allows one to compute the instantaneous position of the oscillator, which is useful in solving specific time-dependent questions.

Amplitude and Maximum Speed Relation

In simple harmonic motion, the maximum speed of the oscillating object is directly proportional to both the amplitude of the oscillation and the angular frequency, as given by the relation v_max = A?. This relation enables one to determine the amplitude when the maximum speed and angular frequency are known.

Period and Angular Frequency

The period of oscillation is the time required for one complete cycle of motion. It is related to the angular frequency by the equation ? = 2?/T, where T is the period. Angular frequency provides a measure of how quickly the system oscillates and is essential for relating time-domain measures to amplitude and phase in the equations of motion.

Simple Harmonic Motion

Simple harmonic motion is the oscillatory motion observed in systems where the restoring force is directly proportional to the displacement from equilibrium and is directed towards that equilibrium position. This concept is crucial in understanding systems such as air-track gliders attached to springs or pendulums, and it forms the basis of the mathematical models that describe periodic motion.

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